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Radius of Gyration Class 11
As a measure of the way in which the mass of a rotating rigid body is distributed with respect to the axis of rotation, we define a new parameter known as the radius of gyration.
It is the distance whose square when multiplied with the mass of the body gives the moment of inertia of the body about the given axis. For a body of mass $M,$ if $K$ is radius of gyration
$\mathrm{I}=\mathrm{MK}^{2}$
$\mathrm{K}=\sqrt{\frac{\mathrm{I}}{\mathrm{M}}} \quad \mathrm{K}=\sqrt{\frac{\sum_{i=1}^{n} m_{i} r_{i}^{2}}{M}}$
Through this concept a real body (particular) is replaced by a point mass for dealing its rotational motion. e.g., in case of a solid sphere rotating about an axis through its centre of mass.
$\mathrm{K}=\sqrt{\frac{\mathrm{I}}{\mathrm{M}}}=\sqrt{\frac{(2 / 5) \mathrm{MR}^{2}}{\mathrm{M}}}=\mathrm{R} \sqrt{\frac{2}{5}}$
So instead of solid sphere we can assume a point mass M at a distance $(\mathrm{R} \sqrt{\frac{2}{5}})$ from the axis of rotation for dealing the rotational motion of the solid sphere.
K depends on
(i) axis of rotation(ii) distribution of mass of body
K does not depend on
(i) Mass of the body $\quad$
(ii) Position of body
(iii) On all angular physical quantity
Angular Momentum
The moment of linear momentum of a moving particle with respect to axis of rotation is known as angular momentum.
It is a vector quantity, which is often represented by $\overrightarrow{\mathrm{L}}$ or $\overrightarrow{\mathrm{J}}$.
It is an axial vector (i.e. always perpendicular to the plane of motion)
$\begin{aligned} \text { Angular momentum } & \vec{J}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{p}} \\ &=\overrightarrow{\mathrm{r}} \times(\mathrm{m} \overrightarrow{\mathrm{v}})=\mathrm{m}(\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{v}}) \end{aligned}$
or $\quad \vec{J}=r p \sin \theta \hat{n}$
$\theta$ is angle between $\vec{r}$ and $\vec{v}$
$\hat{n}$ is unit vector perpendicular to plane of $\overrightarrow{\mathrm{r}}$ and $\overrightarrow{\mathrm{v}}$
The direction of angular momentum is perpendicular to the plane of $\overrightarrow{\mathrm{r}}$ and $\overrightarrow{\mathrm{v}}$ and it given by right hand
screw rule.
$\begin{array}{ll}{\text { If } \theta=0^{\circ},} & {J=0 \quad \text { (Minimum) }} \\ {\text { If } \theta=90^{\circ},} & {J=m \vee r \quad \text { (Maximum) }}\end{array}$
S.I. Unit of $J$ is Jule $\times$ sec (same as that of Planck's const.)
Dimension: $\mathrm{M}^{1} \mathrm{L}^{2} \mathrm{T}^{-1}$
If direction of rotation is anticlockwise, angular momentum is taken positive and if direction of rotation is clockwise, angular momentum is taken negative.
The angular momentum of a system of particles is equal to the vector sum of angular momentum of each particle
$\vec{J}=\vec{J}_{1}+\vec{J}_{2}+\vec{J}_{3}+\ldots \ldots \ldots . \quad \vec{J}=\sum_{i=1}^{n} \vec{J}_{i}$
- radius of gyration does not depends on the mass of rigid body. Graphical relation between log K and log I.
